Abstract

We examine two questions regarding Fourier frequencies for a class of iterated function systems (IFS). These are iteration limits arising from a fixed finite families of affine and contractive mappings in R-d, and the "IFS" refers to such a finite system of transformations, or functions. The iteration limits are pairs (X, mu) where X is a compact subset of R-d (the support of mu), and the measure mu is a probability measure determined uniquely by the initial IFS mappings, and a certain strong invariance axiom. The two questions we study are: (1) existence of an orthogonal Fourier basis in the Hilbert space L-2(X, mu); and (2) explicit constructions of Fourier bases from the given data defining the IFS. (C) 2007 Elsevier Inc. All rights reserved.